Definition:Greatest Common Divisor/Real Numbers

Definition

Let $a, b \in \R$ be commensurable.

Then there exists a greatest element $d \in \R_{>0}$ such that:

$d \divides a$

$d \divides b$

where $d \divides a$ denotes that $d$ is a divisor of $a$.

This is called the greatest common divisor of $a$ and $b$ and denoted $\gcd \set {a, b}$.

The greatest common divisor is often seen abbreviated as GCD, gcd or g.c.d.

Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs.

The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable.

Highest common factor when it occurs, is usually abbreviated as HCF, hcf or h.c.f.

It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$.

The archaic term greatest common measure can also be found, mainly in such as Euclid's The Elements.